Let's solve the problem step-by-step to determine the difference in total account balances between simple interest and monthly compounded interest.
1. Given Values:
- Principal Amount ([tex]\( P \)[/tex]): \[tex]$5900
- Annual Percentage Rate (APR): 2.996% = 0.02996 (as a decimal)
- Time (\( T \)): 15 years
2. Simple Interest Calculation:
The formula for simple interest is:
\[ \text{Simple Interest} = P \times r \times T \]
Where:
- \( P \) is the principal amount
- \( r \) is the annual interest rate (as a decimal)
- \( T \) is the time in years
\[ \text{Simple Interest} = 5900 \times 0.02996 \times 15 \]
\[ \text{Simple Interest} = 2651.46 \]
The total amount with simple interest is:
\[ \text{Total (Simple)} = P + \text{Simple Interest} \]
\[ \text{Total (Simple)} = 5900 + 2651.46 \]
\[ \text{Total (Simple)} = 8551.46 \]
3. Compound Interest Calculation:
The formula for compound interest with monthly compounding is:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount.
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times interest is compounded per year.
- \( t \) is the number of years the money is invested.
For monthly compounding:
- \( n \) = 12 (since interest is compounded monthly)
- \( t \) = 15 years
\[ A = 5900 \left(1 + \frac{0.02996}{12}\right)^{12 \times 15} \]
\[ A = 5900 \left(1 + 0.0024967 \right)^{180} \]
\[ A = 5900 \times 1.056.472... \]
\[ A \approx 9242.31 \]
4. Difference between Compound and Simple Interest Totals:
\[ \text{Difference} = \text{Total (Compound)} - \text{Total (Simple)} \]
\[ \text{Difference} = 9242.31 - 8551.46 \]
\[ \text{Difference} = 690.85 \]
So, the difference in the total account balances if simple interest is applied, compared to monthly compounded interest is approximately:
\[ \boxed{690.85} \]
Therefore, the correct answer is not one of the options given in the question. The closest accurate numerical result to the true difference is $[/tex]\[tex]$ 690.85$[/tex].
